Problem: Simplify; express your answer in exponential form. Assume $q\neq 0, x\neq 0$. $\dfrac{{(q)^{3}}}{{(q^{-4}x^{2})^{2}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${q}$ to the exponent ${3}$ . Now ${1 \times 3 = 3}$ , so ${(q)^{3} = q^{3}}$ In the denominator, we can use the distributive property of exponents. ${(q^{-4}x^{2})^{2} = (q^{-4})^{2}(x^{2})^{2}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(q)^{3}}}{{(q^{-4}x^{2})^{2}}} = \dfrac{{q^{3}}}{{q^{-8}x^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{3}}}{{q^{-8}x^{4}}} = \dfrac{{q^{3}}}{{q^{-8}}} \cdot \dfrac{{1}}{{x^{4}}} = q^{{3} - {(-8)}} \cdot x^{- {4}} = q^{11}x^{-4}$.